QCD I: Introduction, Tree Graph Predictions, and Jets
14.1 The colour degree of freedom
The first intimation of a new, unrevealed degree of freedom of matter came from baryon spectroscopy (Greenberg 1964; see also Han and Nambu 1965, and Tavkhelidze 1965). For a baryon made of three spin-12quarks, the original non-relativistic quark model wave-function took the form
ψ3q=ψ3q,spaceψ3q,spinψ3q,flavour. (14.1) It was soon realized (e.g. Dalitz 1965) that the product of these space, spin and flavour wavefunctions for the ground state baryons wassymmetricunder interchange of any two quarks. For example, the Δ++ state mentioned in section 12.2.3 is made of three u quarks (flavour symmetric) in the JP =
3 2
+ state, which has zero orbital angular momentum and is hence spatially symmetric, and a symmetricS=32 spin wavefunction. But we saw in section 7.2 that quantum field theory requires fermions to obey the exclusion principle – i.e. the wavefunction ψ3q should be antisymmetric with respect to quark interchange. A simple way of implementing this requirement is to suppose that the quarks carry a further degree of freedom, called colour, with respect to which the 3q wavefunction can be antisymmetrized, as follows (Fritzsch and Gell-Mann 1972, Bardeen, Fritzsch and Gell-Mann 1973). We introduce acolour wavefunction with colour indexα:
ψα (α= 1,2,3).
14.1. The colour degree of freedom 75
We are here writing the three labels as ‘1, 2, 3’, but they are often referred to by colour names such as ‘red, blue, green’; it should be understood that this is merely a picturesque way of referring to the three basic states of this degree of freedom, and has nothing to do with real colour! With the addition of this degree of freedom we can certainly form a three-quark wavefunction which is antisymmetric in colour by using the antisymmetric symbol Eαβγ , namely1
ψ3q, colour = Eαβγ ψαψβ ψγ (14.2) and this must then be multiplied into (14.1) to give the full 3q wavefunction.
To date, all known baryon states can be described this way, i.e. the symmetry of the ‘traditional’ space-spin-flavour wavefunction (14.1) is symmetric overall, while the required antisymmetry is restored by the additional factor (14.2). As far as meson (¯qq) states are concerned, what was previously a π+ wavefunction d∗ u is now
1 (d ∗
√ 1u1 + d ∗ 2u2 + d ∗ 3u3) (14.3)
3 √
which we write in general as (1/ 3)d† αuα. We shall shortly see the group theoretical significance of this ‘neutral superposition’, and of (14.2). Mean-while, we note that (14.2) is actually the only way of making an antisymmetric combination of the three ψ’s; it is therefore called a (colour) singlet. It is re-assuring that there is only one way of doing this – otherwise, we would have obtained more baryon states than are physically observed. As we shall see in section 14.2.1, (14.3) is also a colour singlet combination.
The above would seem a somewhat artificial device unless there were some physical consequences of this increase in the number of quark types – and there are. In any process which we can describe in terms of creation or annihilation of quarks, the multiplicity of quark types will enter into the relevant observable cross section or decay rate. For example, at high energies the ratio
σ(e+e− → hadrons)
R = (14.4)
σ(e+e− → μ+μ−)
will, in the quark parton model (see section 9.5), reflect the magnitudes of the individual quark couplings to the photon:
R = E 2
e a (14.5)
a
where a runs over all quark types. For five quarks u, d, s, c, b with respective charges 2 3 , −3 1 , −1 3 3 , 2, −1 3 , this yields
Rno colour = 11 (14.6)
9
1In (14.2) each ψ refers to a different quark, but we have not indicated the quark labels explicitly.
14.1. The colour degree of freedom 77
FIGURE 14.2 τ decay.
FIGURE 14.3
Triangle graph forπ0 decay.
distribution functions consistent with deep inelastic scattering, the parton model gives a good first approximation to the data.
Finally, we mention the rate forπ0 →γγ. As will be discussed in section 18.4, this process is entirely calculable from the graph shown in figure 14.3 (and the one with the γ’s ‘crossed’), where ‘q’ is u or d. The amplitude is proportional to the square of the quark charges, but because the π0 is an isovector, the contributions from the u¯u and d¯d states have opposite signs (see section 12.1.3). Thus the rate contains a factor
((2/3)2−(1/3)2)2= 1
9. (14.9)
However, the original calculation of this rate by Steinberger (1949) used a model in which the proton and neutron replaced the u and d in the loop, in which case the factor corresponding to (14.9) is just 1 (since the n has zero charge). Experimentally the rate agrees well with Steinberger’s calculation, indicating that (14.9) needs to be multiplied by 9, which corresponds toNc= 3 identical amplitudes of the form shown in figure 14.3, as was noted by Bardeen, Fritzsch and Gell-Mann (1973).
76 14. QCD I: Introduction, Tree Graph Predictions, and Jets
FIGURE 14.1
The ratio R (see (14.4)). Figure reprinted with permission from L. Montanet et al. Physical Review D 50 1173 (1994). Copyright 1994 by the American Physical Society.
and 11
Rcolour = (14.7)
3
for the two cases, as we saw in section 9.5. (The valuesR = 2 below the charm threshold, and R = 10/3 below the b threshold, were predicted by Bardeen et al. 1973). The data (figure 14.1) rule out (14.6), and are in good agreement with (14.7) at energies well above the b threshold, and well below the Z0 resonance peak. There is an indication that the data tend to lie above the parton model prediction; this is actually predicted by QCD via higher-order corrections, as will be discussed in section 15.1.
A number of branching fractions also provide simple ways of measuring the number of colours Nc. For example, consider the branching fraction for τ− → e−ν¯eντ (i.e. the ratio of the rate for τ− → e−ν¯eντ to that for all other decays). τ− decays proceed via the weak process shown in figure 14.2, where
e νμ, or ¯
the final fermions can be e−ν¯ , μ− ¯ ud, the last with multiplicity Nc. Thus
B(τ− → e − ¯ 1
νeντ )≈ . (14.8)
2 + Nc Experiments give B ≈ 18 % and hence Nc ≈ 3.
Similarly, the branching fraction B(W− → e−ν¯ ) is e ∼ 3+2N1 c (from f = e, μ, τ,u and c). Experiment gives a value of 10.7 %, so again Nc ≈ 3.
In chapter 9 we also discussed the Drell–Yan process in the quark parton model; it involves the subprocess q¯q → l¯l which is the inverse of the one in (14.4). We mentioned that a factor of 1 3 appears in this case: it arises because we must average over the nine possible initial q¯q combinations (factor 1 ) and then sum over the number of such states that lead to the colour neutral photon, which is 3 (¯q1q1,¯q2q2 and ¯q3q3). With this factor, and using quark
9
76 14. QCD I: Introduction, Tree Graph Predictions, and Jets
FIGURE 14.1
The ratioR(see (14.4)). Figure reprinted with permission from L. Montanet et al. Physical Review D 50 1173 (1994). Copyright 1994 by the American Physical Society.
and
Rcolour= 11
3 (14.7)
for the two cases, as we saw in section 9.5. (The valuesR= 2 below the charm threshold, andR= 10/3 below the b threshold, were predicted by Bardeenet al. 1973). The data (figure 14.1) rule out (14.6), and are in good agreement with (14.7) at energies well above the b threshold, and well below the Z0 resonance peak. There is an indication that the data tend to lieabove the parton model prediction; this is actually predicted by QCD via higher-order corrections, as will be discussed in section 15.1.
A number of branching fractions also provide simple ways of measuring the number of coloursNc. For example, consider the branching fraction for τ−→e−ν¯eντ (i.e. the ratio of the rate forτ−→e−ν¯eντ to that for all other decays). τ− decays proceed via the weak process shown in figure 14.2, where the final fermions can be e−ν¯e, μ−ν¯μ, or ¯ud, the last with multiplicity Nc. Thus
B(τ− →e−ν¯eντ)≈ 1 2 +Nc
. (14.8)
Experiments giveB ≈18 % and henceNc≈3.
Similarly, the branching fraction B(W− →e−ν¯e) is ∼ 3+2N1 c (from f = e, μ, τ,u and c). Experiment gives a value of 10.7 %, so again Nc≈3.
In chapter 9 we also discussed the Drell–Yan process in the quark parton model; it involves the subprocess q¯q →l¯l which is the inverse of the one in (14.4). We mentioned that a factor of 13 appears in this case: it arises because we must average over the nine possible initial q¯q combinations (factor 19) and then sum over the number of such states that lead to the colour neutral photon, which is 3 (¯q1q1,q¯2q2 and ¯q3q3). With this factor, and using quark
14.1. The colour degree of freedom 77
FIGURE 14.2 τ decay.
FIGURE 14.3
Triangle graph for π0 decay.
distribution functions consistent with deep inelastic scattering, the parton model gives a good first approximation to the data.
Finally, we mention the rate for π0 → γγ. As will be discussed in section 18.4, this process is entirely calculable from the graph shown in figure 14.3 (and the one with the γ’s ‘crossed’), where ‘q’ is u or d. The amplitude is proportional to the square of the quark charges, but because the π0 is an isovector, the contributions from the u¯u and d¯d states have opposite signs (see section 12.1.3). Thus the rate contains a factor
((2/3)2 − (1/3)2)2 = 1 . (14.9) 9
However, the original calculation of this rate by Steinberger (1949) used a model in which the proton and neutron replaced the u and d in the loop, in which case the factor corresponding to (14.9) is just 1 (since the n has zero charge). Experimentally the rate agrees well with Steinberger’s calculation, indicating that (14.9) needs to be multiplied by 9, which corresponds to Nc = 3 identical amplitudes of the form shown in figure 14.3, as was noted by Bardeen, Fritzsch and Gell-Mann (1973).
14.2. The dynamics of colour 79 12.2 that antiquarks belong to the complex conjugate of the representation (or multiplet) to which quarks belong. Thus if a quark colour triplet wavefunction ψα transforms under a colour transformation as
ψα→ψα' =Vαβ(1)ψβ (14.10) whereV(1) is a 3×3 unitary matrix appropriate to theT = 1 representation of SU(2) (cf (12.48) and (12.49)), then the wavefunction for the ‘anti’-triplet isψ∗α, which transforms as
ψα∗ →ψ∗'α =Vαβ(1)∗ψ∗β. (14.11) Given this information, we can now construct colour singlet wavefunctions for mesons, built from ¯qq. Consider the quantity (cf (14.3))E
αψ∗αψα whereψ∗ represents the antiquark and ψ the quark. This may be written in matrix notation as ψ†ψ where theψ† as usual denotes the transpose of the complex conjugate of the column vectorψ. Then, taking the transpose of (14.11), we find that ψ† transforms by
ψ† →ψ†'=ψ†V(1)† (14.12) so that the combinationψ†ψ transforms as
ψ†ψ→ψ†'ψ'=ψ†V(1)†V(1)ψ=ψ†ψ (14.13) where the last step follows since V(1) is unitary (compare (12.58)). Thus the product isinvariantunder (14.10) and (14.11) – that is, it is a colour singlet, as required. This is the meaning of the superposition (14.3).
All this may seem fine, but there is a problem. The three-dimensional representation of SU(2)c which we are using here has a very special nature:
the matrixV(1)can be chosen to bereal. This can be understood ‘physically’
if we make use of the great similarity between SU(2) and the group of rota-tions in three dimensions (which is the reason for the geometrical language of isospin ‘rotations’, and so on). We know very well how real three-dimensional vectors transform, namely by an orthogonal 3×3 matrix. It is the same in SU(2). It is always possible to choose the wavefunctionsψto be real, and the transformation matrixV(1) to be real also. SinceV(1) is, in general, unitary, this means that it must be orthogonal. But now the basic difficulty appears:
there is no distinction between ψ and ψ∗! They both transform by the real matrix V(1). This means that we can make SU(2) invariant (colour singlet) combinations for ¯q¯q states, and for qq states, just as well as for ¯qq states – indeed they are formally identical. But such ‘diquark’ (or ‘antidiquark’) states are not found, and hence – by assumption – shouldnot be colour singlets.
The next simplest possibility seems to be that the three colours corre-spond to the components of an SU(3)c triplet. In this case the quark colour wavefunctionψα transforms as (cf (12.74))
ψ→ψ' =Wψ (14.14)
78 14. QCD I: Introduction, Tree Graph Predictions, and Jets